Treebeard's Stumper Answer 10 March 2000

Strange Views

Part of the challenge of technical drawing or drafting is learning to show solid objects in flat drawings. A common method is to use orthographic projections to show the edges of an object with front, top, and side views. Here's an example, and two stumpers. The puzzle is to study the three views and reconstruct the solid objects they show. All surfaces are flat, though not necessarily at right angles to each other. There are no hidden edges. Hint: each object can be carved from a solid block with just one or two straight cuts!

Four Dunn Middle School students were the first to independently find both stumper blocks shown below. This requires real visual thinking! The first object is made with two straight cuts, one diagonally across the top and straight down, the other diagonally across the side and in until they meet. The other object can be made with a single cut from the top diagonal down at an angle to the front corner. It's not surprising that much drafting is now done on computers with CAD software that lets the designer manipulate solid objects in virtual 3D space.

Notes:

Graybear did me one better and produced developments of each object showing how they can be made from a piece of paper. If each square is 1 by 1, then how long are the diagonals? (Hint: c² = a² + b²)

Challenge accepted, Graybear! I cut and pasted your developments into a graphics program, enlarged them and added glue tabs, printed them on heavy paper with my ink jet printer, and made them. Nice fit!

My plans are available here and here. Save the images and load them into a graphics program like Paint Shop Pro. Print them full page in landscape mode on card stock or photo paper. Crease all the folds (scoring will help) and paste them together with a glue stick. Let the ink dry a long time before folding so they won't be as smudged as mine! This is a handy way to make paper model templates for a drafting class. Now there's an idea for a useful Web site!

I didn't think I'd have anything to say about this stumper, but I was wrong. I presented this stumper in each of my science classes at Dunn Middle School, and gave the kids just five minutes to do it. Four kids (of 48) got both stumpers. What's interesting is that three of the four kids are in special tutoring classes. We all learn in different ways, and this shows that kids who need help with normal school tasks can excel at other important tasks.

I've always been skeptical about left-brain / right-brain dichotomies. That dualism just seems too simple and archaic to be anything more than a metaphor. But it's undeniable that we're different, and it's a step in the right direction for teachers.

In this election year, it seems politicians can't talk about education without mentioning "standardized testing." That scares me. I don't see the point of trying to make everyone the same. This policy hurts kids, and it will hurt our country in the long run. I want diversity and more opportunities for kids in schools, not narrow standards. Arts and vocational training and other educational alternatives are important. They allow different kids to shine.

Of course that's just my opinion. I may be wrong.

Here are some Web links for further research on visual learning:

• My Getting Around (30 April 1999) and Twisted Wires (18 April 1997) stumpers are two more puzzles that require visual thinking.

• Maths Primer for the Card Modeler has practical help on building 3D paper models and visualizing orthographic projections.

• I found a site in Australia that discusses the use of 3D CAD software in teaching engineering drafting as a visualization aide. They find pluses and minuses. Read An Evaluation of Curriculum Changes in Engineering Graphics and A Study on Computer-Based Geometric Modelling in Engineering Graphics, by Zlatko Gradinscak and others.

• There's lots of information on the Web about different learning modalities. The Penn State EDUC 520 Learning Theory in the Classroom site has lots of links, and so does About.com.

• Learning to Learn has several online multiple-choice tests based on work by Howard Gardner that you can try to find your own teaching/learning style. My results were pretty ambiguous. (That's good!)

• PBS has a disturbing report on the use of Ritalin and other drugs by children for treatment of Attention Deficit Disorder (ADHD). This is an extreme way to enforce standards in schools. On the other hand, I know how I depend on coffee.

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